Solve for $x$ : $ 5|x - 8| - 2 = -6|x - 8| + 3 $
Answer: Add $ {6|x - 8|} $ to both sides: $ \begin{eqnarray} 5|x - 8| - 2 &=& -6|x - 8| + 3 \\ \\ { + 6|x - 8|} && { + 6|x - 8|} \\ \\ 11|x - 8| - 2 &=& 3 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 11|x - 8| - 2 &=& 3 \\ \\ { + 2} &=& { + 2} \\ \\ 11|x - 8| &=& 5 \end{eqnarray} $ Divide both sides by ${11}$ $ \dfrac{11|x - 8|} {{11}} = \dfrac{5} {{11}} $ Simplify: $ |x - 8| = \dfrac{5}{11}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -\dfrac{5}{11} $ or $ x - 8 = \dfrac{5}{11} $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -\dfrac{5}{11} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -\dfrac{5}{11} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -\dfrac{5}{11} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $11$ $ x = - \dfrac{5}{11} {+ \dfrac{88}{11}} $ $ x = \dfrac{83}{11} $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = \dfrac{5}{11} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& \dfrac{5}{11} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& \dfrac{5}{11} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $11$ $ x = \dfrac{5}{11} {+ \dfrac{88}{11}} $ $ x = \dfrac{93}{11} $ Thus, the correct answer is $x = \dfrac{83}{11} $ or $x = \dfrac{93}{11} $.